\magnification=\magstep1 \baselineskip=15pt
\def \Box {\hbox{}\nobreak \vrule width 1.6mm height 1.6mm depth 0mm
\par \goodbreak \smallskip}  \vskip .5cm

\centerline{ \bf FOUR COUNTEREXAMPLES \ IN}
\centerline{\bf COMBINATORIAL ALGEBRAIC GEOMETRY}

\vskip .7cm

{\baselineskip=13pt
\centerline{ \bf Bernd Sturmfels}
\centerline{ Department of Mathematics}
\centerline{ University of California}
\centerline{ Berkeley, CA 94720, U.S.A.}
\centerline{ \tt bernd@math.berkeley.edu}
}
\vskip 1.2cm

\noindent 
We present counterexamples to four conjectures which 
appeared in the literature in commutative algebra and algebraic geometry.
The four questions to be studied are largely unrelated, and yet our answers 
are connected by a common thread: they are combinatorial
in nature, involving monomial ideals and binomial ideals,
and they were found by exhaustive computer search using the
symbolic algebra systems { \tt  Maple } and { \tt Macaulay 2}.

In Section 1 we answer Chandler's question [4, Question 1] whether
the Castelnuovo-Mumford regularity of a homogeneous polynomial ideal $I$
satisfies the inequality $\,reg(I^r) \leq r \cdot reg(I)$.
We present a characteristic-free counterexample
generated by only eight monomials;
this improves an earlier example by Terai [5, Remark 3].
In Section 2 we settle a conjecture published 
two decades ago by Brian\c con and 
Iarrobino [2, page 544], by showing that the most singular point on the 
Hilbert scheme of points need not be the monomial ideal with most generators.
In Section 3 we construct a smooth projectively normal curve which is
defined by quadrics but is not Koszul; this solves a problem posed
by Butler [3, Problem 6.5]
and Polishchuk [16, page 123]. Section 4 
disproves an overly optimistic conjecture of mine
[23,  Example 13.17] about the
Gr\"obner bases of a certain toric $4$-fold.

Each of the four counterexamples is displayed in the user language of
{\tt Macaulay 2}; see [11]. We encourage the readers to
try out these lines of code, and to enjoy their own explorations
in combinatorial algebraic geometry. Naturally,
our results raise many more questions than they answer,  and
several new open problems will be stated in this article.



\beginsection 1. Regularity of powers of ideals

The {\sl Castelnuovo-Mumford regularity} $reg(I)$ of a homogeneous ideal 
$I$ in $k[x_1,\ldots,x_n]$ is the maximum of 
total degree minus homological degree for any minimal syzygy of $I$.
Chandler [4] raised the question whether the following inequality always
holds,
$$ reg(I^r) \quad \leq \quad r \cdot reg(I) ,  \eqno (1.1) $$
and she proves this when $k[x_1,\ldots,x_n]/I$
has Krull dimension $\leq 1$. The same result was obtained
by Geramita, Gimigliano and Pitteloud in [10]. Smith-Swanson [20] and 
Hoa-Trung [12] investigate Chandler's problem for monomial ideals, and 
they provide upper bounds 
for $reg(I^r)$ which are specific to the monomial case.

Kodiyalam [14]  shows that $\,  reg(I^r) \, \leq \,
r \cdot reg(I) \, + \, c_I \,$,
where $c_I$ is a constant depending only on $I$,  and he notes
that (1.1) holds if and only if all powers of an
ideal with a linear resolution also have a linear resolution 
[14, Remark 2]; see also [7].
(Recall that $I$ has a {\it linear resolution}
if $I$ is generated by homogeneous polynomials of degree equal to $reg(I)$.)

It was first observed by Terai [5, Remark 3]
that the Stanley-Reisner ideal (cf.~[22]) of the minimal
triangulation of the real projective plane violates
(1.1) in characteristic $\not= 2$.
We present this monomial ideal and the asserted property in 
the \ {\tt Macaulay 2} \ language:


\vskip .2cm

{ \baselineskip=12pt \tt
i1 : R = QQ[a,b,c,d,e,f];

\vskip .1cm

i2 : M = ideal(a*b*c,a*b*f,a*c*e,a*d*e,a*d*f, \hfill \break
\phantom{dadadadaddadaadadadadada}       b*c*d,b*d*e,b*e*f,c*d*f,c*e*f);

i3 : betti res M     \hfill \break
%    total: 10 15 6 \hfill \break
    \phantom{tota}3: 10 15 6   }

\vskip .1cm

{ \baselineskip=12pt \tt
i4 : betti res M\^{$\,\!$}2      \hfill \break
% i4 : betti res M\^{ }2      \hfill \break
\phantom{tot}        6: 55 144 150 80 21 .     \hfill \break
\phantom{tot}        7:  . \  . \  .  .  .  1   
}


\vskip .1cm

\noindent
These two tables of  Betti numbers show that $reg(M) = 3$ but  $reg(M^2) = 7$.
The disadvantage of the projective plane example is that 
it does not work if the characteristic
of $k$ is  $2$, since $reg(M) = 4$ in this case.
Note how the tables in the output change when
the field of rational numbers ${\tt QQ}$ is replaced by 
the two-element field ${\tt ZZ/2}$ in the input line {\tt i1}.

The following is the main result in this section.

\proclaim Theorem 1.1.
Let $M$ be an ideal generated by $r$ monomials in $k[x_1,\ldots,x_n]$.
\item{(a)} If $r \leq 7$ and $M$ has a linear resolution, then
its square $M^2$ has a linear resolution.
\item{(b)} Statement (a) is false for $r=8$, $n=6$
and any field $k$.

\noindent {\sl Sketch of Proof: }
A monomial ideal has a linear resolution if and only if its
polarization has a linear resolution, so we may assume that 
$M$ is a square-free monomial ideal. By the Eagon-Reiner Theorem [8], 
$M$ has a linear resolution if and only if
$M$ equals the Stanley-Reisner ideal $I_{\Delta^*}$
of the Alexander dual $\Delta^*$ of
a Cohen-Macaulay simplicial complex $\Delta$ on $\{1,\ldots,n\}$.
In this context, {\it Alexander duality} means that $M$ is generated by
the square-free monomials $\prod_{i \not\in \sigma} x_i$ where 
$\sigma$ runs over all facets of $\Delta$.

Part (a) of Theorem 1.1 was proved by investigating
all Cohen-Macaulay complexes with $r \leq 7$ facets. 
We constructed these complexes
by exhaustively enumerating all strongly connected 
complexes having $r \leq 7$ facets which are not cones.
The dimension of such a complex is at most $5$.
(For instance, in dimension $1$ this amounts to listing
all connected graphs on $\leq 7$ vertices which are not trees.)
This enumeration was done in {\tt Maple}. Numerous tricks 
and reductions were used to contain the combinatorial explosion.
As a byproduct we found that every Cohen-Macaulay complex with
$r \leq 7$ facets is shellable.

The counterexample in part (b) is given by the 
$2$-dimensional simplicial complex $\, \Delta = \bigl\{ 
123,124,125,126,134,156, 245,236 \bigr\}$.
The eight triangles are listed in a shelling order.
This complex is best visualized by
drawing the octahedron with vertices
$\,(1,0,0),(-1,0,0)$, $ (0,1,0),(0,0,1),(0,-1,0),(0,0,-1)$,
labelled $1,2,3,4,5,6$ in this order.
The four tetrahedra $\,1234,\,1245,\,1256\,$ and $\,1236\,$
define a triangulation of this octahedron
which contains $\Delta$ as a subcomplex.
Here is the corresponding ideal $M = I_{\Delta^*}$
in {\tt Macaulay 2} format:

\vskip .2cm

{ \baselineskip=12pt \tt

i5 : M = ideal(d*e*f,c*e*f,c*d*f,c*d*e,b*e*f,b*c*d,a*c*f,a*d*e);

\vskip .1cm

i6 : betti res M   \hfill \break
%    total: 8 11 4  \hfill \break
\phantom{tot}         3: 8 11 4   

\vskip .2cm

i7 : betti res M\^{$\,\!$}2  \hfill \break
%    total: 36 85 79 38 10 1 \hfill \break
\phantom{tot}         6: 36 84 75 32 \ 6  . \hfill \break
\phantom{tot}         7: .  \ 1 \ 4 \  6 \  4   1

\vskip .1cm

}

\noindent
Since $\Delta$ is shellable, it is Cohen-Macaulay over every field
[22, Theorem III.2.5]. Hence
the ideal $M$ has a linear resolution over every field.
On the other hand, consider the $85$ minimal first syzygies of 
its square $M^2$. They are field-independent and one of them is non-linear:
it is the first syzygy between $a^2 cdef$ and $b^2 cdef$.
In the notation of Cutkosky, Herzog and Trung [7, \S 3],
our example satisfies $\, reg_1(M) = 3 \,$ and $\,reg_1(M^2) = 7$. \Box

\vskip .1cm

The prototype of a shellable simplicial complex $\Delta$
is the boundary complex $\Delta = \partial P$ 
of a simplicial convex polytope $P$. The corresponding 
monomial ideal $\,M = I_{\Delta^*}\,$ plays a prominent role
in toric geometry, namely, following Cox [6, Theorem 3.7],
it is the irrelevant ideal in the {\it homogeneous
coordinate ring} of the toric variety $X_P$ associated with $P$.
The a linear resolution of $M$ is essentially the coboundary complex 
of the polytope $P$, and the explicit form of this resolution is
useful for computing cohomology of 
sheaves on $X_P$.  It is natural to ask what can be said about
the Koszul homology of powers of the ideal $M$.

\proclaim Problem 1.2.
Let $P$ be a simplicial polytope and $M$ the Stanley-Reisner ideal
of the Alexander dual of $\partial P$. Does the square $M^2$ of
this ideal have a linear resolution~?



\beginsection 2. The most singular point on the Hilbert scheme of points

Which is the ``most singular'' point on the Hilbert scheme
$\,Hilb^d(S)$ of colength $d$ ideals in the polynomial ring
$\,S = k[x_1,\ldots,x_n]$~?
In other words, which artinian ideal $I$ of colength $d$ in $S$
has the tangent space $\,Hom_S(I,S/I)\,$ of maximal $k$-dimension~?
As usual in the study of Hilbert schemes, it suffices to consider
Borel-fixed monomial ideals $I$, for which the problem of determining
$\,dim_k Hom_S(I,S/I)\,$ is a purely combinatorial one.

Brian\c{c}on and Iarrobino  [2, page 544]
made a precise conjecture which would answer the above question.
A special version of their conjecture states that
powers of the maximal ideal $M = \langle x_1,\ldots,x_n \rangle$ 
are always most singular in their own Hilbert scheme:

\proclaim Conjecture 2.1. {\rm (Brian\c{c}on and Iarrobino [2]) }
If $d = {n+s-1 \choose n}$ and $\,I \in Hilb^d(S)\,$ then
$$ dim_k Hom_S(I,S/I) \quad \leq \quad
dim_k Hom_S(M^s,S/M^s) .$$

The number on the right hand side equals
${n+s-1 \choose n-1}{n+s-2 \choose n-1}$ because
$Hom_S(M^s,S/M^s) \,$ is isomorphic as a $k$-vector space to
$\,Hom_k(M^s/M^{s+1}, M^{s-1}/M^s)$.

We now state the conjecture of [2] in its general form,
that is, for an arbitrary positive integer $d$. Given $d$, we
choose $s$ such that
$ {n+s-2 \choose n} < d \leq {n+s-1 \choose n}$, and we
let $I(d)$ denote the ideal generated by $M^s$  together with
the $\,{n+s-1 \choose n}-  d \,$
lexicographically highest monomials of degree $s-1$.
It was shown by Berman [1] that $I(d)$ has the
largest number of minimal generators among all monomial ideals
of colength $d$ in $S$.

\proclaim Conjecture 2.2. {\rm (Brian\c{c}on and Iarrobino [2])  }
For all $\,J \in Hilb^d(S)\,$ we have
$$ dim_k Hom_S(J,S/J) \quad \leq \quad
dim_k Hom_S \bigl(I(d),S/I(d) \bigr) .$$

For $n=2$ variables these conjectures are trivially true
since $Hilb^d(k[x,y])$ is smooth. It is hence natural to
explore their validity for $n=3$. The number of monomial
ideals of colength $d$ in $k[x,y,z]$ is the 
coefficient of $t^d$ in MacMahon's classical generating function
$$ 
\prod_{\nu=1}^\infty {1 \over (1-t^\nu)^\nu}  \quad = \quad
1+t+3\,t^2+6\,t^3+13 \,t^4+24 \,t^5+48 \,t^6  + 86 \,t^7 + 
160 \,t^8 
+ \cdots
.$$
See [21, Corollary 18.2] for a proof and further
information on enumerating artinian monomial ideals
in three variables.
For instance, there are $160$ monomial ideals of colength
$8$ in $k[x,y,z]$. These $160$ ideals come in
$33$ types modulo permutations of the three variables,
and of these $33$ types only $12$ are Borel-fixed.
(Here assume $char(k)=0$.) We display the
list of all $12$ colength $8$ 
Borel-fixed ideals in $k[x,y,z]$ in
{\tt Macaulay 2} notation:

\vskip .3cm

{ \baselineskip=12pt \tt
i1 : S = QQ[x,y,z];

\vskip .1cm

i2 : Ideals = $\{$ ideal(x, y, z\^{$\,\!$}8),
        ideal(y*z, x, y\^{$\,\!$}2, z\^{$\,\!$}7),

\quad        ideal(y*z\^{$\,\!$}2, x, y\^{$\,\!$}2, z\^{$\,\!$}6), \qquad
        ideal(y*z\^{$\,\!$}3, x, y\^{$\,\!$}2, z\^{$\,\!$}5),

\quad        ideal(y*z\^{$\,\!$}2, y\^{$\,\!$}2*z, x, y\^{$\,\!$}3, z\^{$\,\!$}5),
        ideal(y\^{$\,\!$}2*z, y*z\^{$\,\!$}3, x, y\^{$\,\!$}3, z\^{$\,\!$}4),

 \quad        ideal(x*y,x*z,y*z\^{$\,\!$}2,x\^{$\,\!$}2,y\^{$\,\!$}2,z\^{$\,\!$}5),
        ideal(x*y,x*z,y*z\^{$\,\!$}3,x\^{$\,\!$}2,y\^{$\,\!$}2,z\^{$\,\!$}4),

\quad        ideal(x*y,x*z\^{$\,\!$}2,y*z\^{$\,\!$}2,x\^{$\,\!$}2,y\^{$\,\!$}2,z\^{$\,\!$}4),
        ideal(x*y,y*z,x*z,x\^{$\,\!$}2,y\^{$\,\!$}2,z\^{$\,\!$}6),

\quad        ideal(x*y,x*z,y*z\^{$\,\!$}2,y\^{$\,\!$}2*z,x\^{$\,\!$}2,y\^{$\,\!$}3,z\^{$\,\!$}4),

 \quad      ideal(x*y,x*z\^{$\,\!$}2,y*z\^{$\,\!$}2,y\^{$\,\!$}2*z,x\^{$\,\!$}2,z\^{$\,\!$}3,y\^{$\,\!$}3)
$\}$ ;

\quad \phantom{ dada} }

We next define a function 
``{\tt hilbtan}'' which computes the
$S$-module $Hom_S(M, S/M)$ and returns a list of four elements:
the number of generators of $M$, the colength of $M$, 
the vector space dimension of $Hom_S(M, S/M)$, 
and the list of generators of $M$:

\vskip .3cm

{ \baselineskip=12pt \tt

i3 : hilbtan = (M) -> (

% \quad \qquad     Tangentspace = Hom( coker( (res M).dd\_{ }2) , R\^{$\,\!$}1/M );
\quad \qquad      Tangentspace = Hom(module M, S\^{$\,\!$}1/M);


\quad \qquad  << $\{$ 
numgens(M), degree(M), degree(Tangentspace), M $\}$ <<endl;) ;

\quad \phantom{ dada} }


\noindent Now we apply our function
``{\tt hilbtan}''  to each of the 
$12$ Borel-fixed ideals listed above:

\vskip .3cm

{ \baselineskip=12pt \tt

i4 : apply(Ideals, hilbtan)

\vskip .1cm

\qquad \qquad $\{$ 3,8,24,ideal | x y z8 | $\}$

\qquad \qquad $\{$ 4,8,24,ideal | yz x y2 z7 | $\}$

\qquad \qquad $\{$ 4,8,24,ideal | yz2 x y2 z6 | $\}$

\qquad \qquad $\{$ 4,8,24,ideal | yz3 x y2 z5 | $\}$

\qquad \qquad $\{$ 5,8,24,ideal | yz2 y2z x y3 z5 | $\}$

\qquad \qquad $\{$ 5,8,24,ideal | y2z yz3 x y3 z4 | $\}$

\qquad \qquad $\{$ 6,8,30,ideal | xy xz yz2 x2 y2 z5 | $\}$

\qquad \qquad $\{$ 6,8,30,ideal | xy xz yz3 x2 y2 z4 | $\}$

\qquad \qquad $\{$ 6,8,36,ideal | xy xz2 yz2 x2 y2 z4 | $\}$
\qquad $\Longleftarrow $

\qquad \qquad $\{$ 6,8,30,ideal | xy yz xz x2 y2 z6 | $\}$

\qquad \qquad $\{$ 7,8,32,ideal | xy xz yz2 y2z x2 y3 z4 | $\}$

\qquad \qquad $\{$ 7,8,32,ideal | xy xz2 yz2 y2z x2 z3 y3 | $\}$

\quad \phantom{dada} }

\noindent
The ideal with the largest-dimensional tangent space is marked
with an arrow. It is
$$ J \quad = \quad
\langle x,y,z^2 \rangle^2 \quad = \quad
\langle x^2, xy, y^2,  x z^2 ,y z^2, z^4 \rangle. $$
This ideal has six minimal generators and its tangent space
$\, Hom_S (J,S/J)\,$ has $k$-dimension $36$.
Berman's lexicographic ideal $I(8)$ appears last in the above output. 
Note that
$$ I(8) \quad = \quad
\langle x^2, xy \rangle \,+ \, M^3 \quad= \quad
\langle x^2, x y,  x z^2, y^3, y^2 z , y z^2, z^3 \rangle. $$
has seven minimal generators but its tangent space
$ Hom_S \bigl(I(8),S/I(8) \bigr)$ has $k$-dimension only $32$.
Our experiment demonstrates that the most singular point
on the Hilbert scheme $Hilb^8 ( k[x,y,z])$ is not the ideal
with the most generators. We summarize our results.

\proclaim Theorem 2.3. Let $n=3$. Then Conjecture 2.2
does not hold for $d=8$ and $d=16$, but it does hold for
all other values of $d$ between $1$ and  $22$.

To get a feeling for how the dimension of the tangent space grows with $d$,
here are some data on Berman's lexicographic ideals.
The first row is the colength $d$, the second row
is the number of generators of $I(d)$ and the
third row is $\,dim_k Hom_S \bigl(I(d),S/I(d) \bigr) $:
$$ \matrix{ \!
    1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17
    & 18 & 19 & \! 20 \cr
\!    3 & 3 & 4 & 6 & 6 & 6 & 7 & 7 & 8 & 10 & 10 & 10 & 10 & 11 & 11 & 11 & 12
    & 12 & 13 & \! 15  \cr
  \! 3 & 6 & 9 & 18 & 21 & 24 & 29 & \underline{32} 
& 43 & 60 & 63 & 66 & 69 & 78 & 81 & \underline{84}
    & 99 & 102 & 123 & \! 150 \cr}
$$
$$
\matrix{
 \! 21 & \! 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 & 31 & 32
    & 33 & 34 & 35 & 36 \cr
\! 15 & \! 15 & 15 & 15 & 16 & 16 & 16 & 16 & 17 & 17 & 17 & 18
    & 18 & 19 & 21 & 21 \cr
\! 153 & \! 156 & 159 & 162 & 177 & 180 & 183 & 186 &
    207 & 210 & 213 & 240 & 243 & 276 & 315 & 318 \cr}
$$

\vskip .1cm

The unique counterexample to Conjecture 2.2 for $d=16$ is the 
following ideal:
$$ \langle  \,x^3,\, x^2 y,\, xyz,\, x^2 z,\, x z^2,\,
 x y^3,\, y^3 z,\, y^2 z^2 , \, y z^3,\, z^4,\, y^5 \,\rangle. $$
The tangent space of this ideal has dimension $88$.

\vskip .1cm

Conjecture 2.1 remains open and has been confirmed
by all our experiments so far. Here are some other
problems, perhaps in order of increasing difficulty:

\vskip .1cm

\proclaim Problems 2.4.
\item{(a)} Is the maximum dimension of the tangent
space at $Hilb^d(k[x,y,z])$ attained at an initial monomial ideal of the
{\sl generic} configuration of $d$ points affine $3$-space~?
(There are far fewer such generic initial ideals than
there are Borel-fixed ideals).
\item{(b)} What is the smallest integer $d$ such
that $Hilb^d(k[x,y,z])$ is reducible~?
Iarrobino [13] shows that $d \leq 102$,
but probably the answer is much smaller. Between $10$ and $20$~?
\item{(c)} Give an exact formula for the dimension of $Hilb^d(k[x,y,z])$,
or, at least, sharpen the asymptotic result in [2] for
the special case of three variables.


Another point that deserves better understanding is the relationship 
between the singularities of the Hilbert scheme of points and
those of the Hilbert scheme of saturated ideals with fixed
Hilbert polynomial. For instance, Reeves and Stillman  [17] 
showed that, in the latter setting, the lexicographic segment ideal 
defines a smooth point. How does this relate to the highly
singular behavior of Berman's lexicographic ideals $I(d)$~?

\beginsection 3. A non-Koszul curve defined by quadrics

In this section we prove the following result:

\proclaim Theorem 3.1. There exists a smooth projectively normal curve of 
genus $7$ in complex projective $5$-space whose coordinate ring is presented 
by  quadrics but is not Koszul.

This answers a question which was asked by Butler [3, Problem 6.5]
and Polishchuk [16, page 123]. I am grateful to Burt Totaro for telling
me about these references and how they might be related to [18].
Our counterexample is a linear section of 
a certain remarkable toric variety which was found 
independently (and for different purposes) by three sets 
of authors:  Schenck and Stillman [19],  Ohsugi and Hibi 
[15, Example 2.2], Roos and myself [18, Example 3].
In [19] this curve is used to disprove a generalization due to Eisenbud
of Green's conjecture on the Koszul homology of canonical curves.

We shall  present the proof of Theorem 3.1 in
the format of a  {\tt Macaulay 2} session.
The vanishing prime ideal of our
curve is generated by five quadrics in six variables:

\vskip .3cm

{ \baselineskip=12pt \tt
i1 : S = QQ[a,b,c,d,e,f]; 

\vskip .2cm

i2 : I = ideal(     a*(c+d) - b*f,

\quad \qquad \qquad \qquad \qquad     b*(b+f) - c*e,

\quad\qquad \qquad \qquad \qquad     c*(a+c+f) - d*(c+d),

\quad\qquad \qquad \qquad \qquad     d*f - e*(b+f),

\quad\qquad \qquad \qquad \qquad     e*e - a*(a+c+f) );

i3 : codim I

o3 = 4

\vskip .2cm


i4 : pdim coker gens I

o4 = 4

\quad \phantom{dada}
}

\noindent
In lines {\tt i3} and {\tt i4} we are verifying that
the ideal $I$ cuts out a curve in $P^5$,
and that this curve is arithmetically Cohen-Macaulay.
To check that it is irreducible, use
the command {\tt decompose(I)}. In the current
version of {\tt Macaulay 2} this works only
over finite fields (such as {\tt ZZ/31991}). This is good enough here:
if a projective curve over ${\bf Q}$ is irreducible and
reduced (try {\tt radical(I)})
in some finite characteristic, then it is also
is irreducible and reduced in characteristic $0$.
Michael Stillman informed me that the command
{\tt decompose(I)} will be available for the
rationals {\tt QQ} in a future version of {\tt Macaulay 2}.

We next check that the projective curve defined by $I$ is non-singular:

\vskip .3cm

{ \baselineskip=12pt \tt
i5 : J = trim( minors(4,jacobian(I)) + I );

i6 : codim J

o6 = 6

\quad \phantom{dada}
}

\noindent
Indeed, the ideal $J$ defining the singular locus 
of $I$ has codimension $6$,
hence defines the empty subset of $P^5$. At this point we
know that the curve is smooth and arithmetically Cohen-Macaulay.
We can therefore conclude that it is projectively normal,
i.e., its coordinate ring $S/I$ is a normal domain.
Moreover, we can now determine the genus of our curve. It is
seven, which is one plus the negated constant term of the Hilbert polynomial:



\vskip .3cm

{ \baselineskip=12pt \tt
i7 : hilbertPolynomial(coker gens I, Projective => false)

o7 = 11 \$i - 6

o7 : QQ[\$i]

\quad \phantom{dada}}

\noindent 
To complete the proof of Theorem 3.1, we must show that the coordinate ring
$R = S/I$ is not a Koszul algebra, i.e., that the resolution of
the residue field $k$ over $R$ is not linear.
We do this by computing the (infinite) resolution of $k$ 
over $R$ up to homological degree $5$:

\vskip .3cm

{ \baselineskip=12pt \tt

i8 : R = S/I

o8 = R

o8 : QuotientRing

\vskip .2cm

i9 : betti res(coker vars R,LengthLimit => 5)

\vskip .1cm

    total: 1 \ 6 \ 20 52 122 281  \hfill \break \phantom{iff} \qquad
        0: 1 \ 6 \ 20 51 111 216 \hfill \break  \phantom{iff} \qquad
        1: . . .  \phantom{r}1 \ 11 \ 65  \hfill \break  \phantom{iff} \qquad
        2: . . .  \phantom{r}. \  . \  .  

\phantom{dada}
}

\noindent
There exists a non-linear syzygy in homological degree $3$;
hence $R=S/I$ is not Koszul. \Box

\vskip .2cm

The following result describes
all the Betti numbers of the infinite resolution above:

\proclaim Proposition 3.2.
The Poincar\'e-Betti series of the graded $k$-algebra $R=S/I$ equals
$$
\sum_{i,j \geq 0}\!
dim_k Tor_i^R(k,k)_j \cdot  t^i \cdot  y^j \,\,\, = \,\,\,
\cases{
{ (1+y t)^2 (1+y^3 t^3)
\over 4 y^5 t^5-y^6 t^6-y^5 t^4-5 y^4 t^4-y^4 t^3+5 y^2 t^2
-4 y t+1} &  if $char(k) =2$, \cr
\phantom{dadadada}
(1+y t)^2 / p(y,t) & if $char(k) \not=2$, \cr
}
$$
where  $
p(y,t) \, = \,
y^9 t^9+y^9 t^8-y^6 t^6-y^6 t^5-y^5 t^5-y^5 t^4-y^4 t^4-4 y^8 t^7+5 
y^7 t^7+5 y^7 t^6+5 y^2 t^2-y^4 t^3-y^3 t^3-4 y^8 t^8-4 y t+1
$.


Both  series begin like
$ \, 1 \,+\, 6 y t \,+\, 20 y^2 t^2\, +\, (51 y^3 + y^4) t^3 + \cdots $.
See the table generated in line {\tt i9} above.
The  characteristic-dependence
appears in homological degree $\geq 4$.
Proposition 3.2 is due to Jan-Erik Roos and was obtained
using the methods in [18, \S 3]. These formulas were originally
included in [18, Example 3] but did not appear in print
because of the publisher's space limitations.
In view of Theorem 3.1, Proposition 3.2,
and the main result of [18], the following variant
of the Butler-Polishchuk question is natural:

\proclaim Problem 3.3.
Does there exist a smooth projectively normal curve  whose
ideal is generated by quadrics but whose Poincar\'e-Betti series
is not a rational function~?

\vskip .2cm

The toric variety of which our curve is a linear section
is gotten by replacing the linear factors
{\tt (c+d)}, {\tt (b+f)}, and {\tt (a+c+f)} in the
five ideal generators 
by new indeterminates, say, {\tt g}, {\tt h} and {\tt i}.
The resulting five quadrics in $9$ variables define
a normal toric $4$-fold of degree  $11$ in $P^8$.
This toric variety was constructed using  graph-theoretic 
methods  by Ohsugi and Hibi in [15]. Note that this
variety is generated by quadrics but has no quadratic 
Gr\"obner basis, since having a quadratic 
Gr\"obner basis implies the Koszul property.

The Gr\"obner bases of another interesting
toric $4$-fold will be studied in the next section. We close this
section with an informal question which is  admittedly provocative.

The counterexample constructed in Theorem 3.1
seems to be quite exceptional in the following sense.
There are many classical varieties in projective geometry
(Segre, Veronese, scrolls, etc.) and representation theory
(Grassmannians, Schubert varieties, etc.) which
are defined by quadratic equations. {\sl All of these
classical varieties possess natural quadratic Gr\"obner bases
in their natural coordinate system.}
Moreover, Eisenbud, Reeves and Totaro [9]
showed that every subscheme of projective space has a
quadratic Gr\"obner basis in a suitable power
of the given embedding, in fact, in the natural coordinates.
This suggests that the homological
property ``Koszul'' may be of less importance
for algebraic geometry than the authors of [3] and [16] (and many others)
have surmised.  Is the algorithmic
property ``has a quadratic Gr\"obner basis''
geometrically more meaningful than Koszulness~?


\beginsection 4. Initial ideals of projectively normal toric varieties

In this section we discuss the following problem.

\proclaim Question 4.1. 
Let $X$ be a projectively normal toric variety in 
projective $n$-space $P^n$
and let $I_X$ be its defining binomial prime ideal
in $S = k[x_0,\ldots,x_n]$. Does there always exist a term order
$\prec$ on $S$ such that the initial monomial ideal
$in_\prec(I_X)$ is Cohen-Macaulay~?

This question makes sense because $S/I_X$ is a Cohen-Macaulay ring,
by Hochster's Theorem [22, Corollary I.7.6].
By the Bayer-Stillman Theorem,  the answer to 
Question 4.1 would be ``yes'' if we were allowed to subject $I_X$ to a generic
linear change of coordinates (suppose the field $k$ is infinite).
But such a coordinate transformation destroys the
binomial structure of the ideal $I_X$, and 
we certainly do not allow such things in the toric context.

Let $d=dim(X)$. The ideal $I_X$ consists of the algebraic relations
among $n+1$ monomials in $d+1$ variables which have the same
total degree. Let ${\cal A} = \{ a_0, a_1,\ldots,a_n\} \subset 
{\bf N}^{d+1}$ be the exponent vectors of these monomials.
The convex hull of ${\cal A}$ in ${\bf R}^{d+1}$ is the polytope underlying
the projective toric variety $X$. The answer to Question 4.1
is also ``yes'' if the configuration ${\cal A}$ admits a unimodular 
regular triangulation; see [23, Theorem 8.3 and Corollary 8.9].
A unimodular regular triangulation $\Delta$ of ${\cal A}$ gives rise to
a term order $\prec$ such that $in_\prec(I_X)$ 
coincides with the Stanley-Reisner ideal of $\Delta$,
which is a shellable ball and hence Cohen-Macaulay by
[22, Theorem III.2.5].

Question 4.1 therefore concerns those projective toric
varieties whose underlying point configuration ${\cal A}$ has no
regular unimodular triangulation. Such a configuration
was given in [23, Example 13.17]
for $d=4, n=8$. We reproduce
it here in {\tt Macaulay 2} notation:

\vskip .3cm

{ \baselineskip=12pt \tt

i1 : R = QQ[v,w,x,y,z];

i2 : S = QQ[a,b,c,d,e,f,g,h,i];

\vskip .1cm

i3 : p = map(R,S, $\{$ z, v*z, w*z, x*z, v*w*x*y*z, v*w*x\^{$\,\!$}2*y\^{$\,\!$}2*z,
\hfill \break
\phantom{dadadada} \qquad\qquad\qquad
v*w\^{$\,\!$}2*x\^{$\,\!$}2*y\^{$\,\!$}3*z, v*w\^{$\,\!$}2*x\^{$\,\!$}3*y\^{$\,\!$}4*z, v*w\^{$\,\!$}2*x\^{$\,\!$}3*y\^{$\,\!$}5*z $\}$);

o3 : RingMap R <--- S

\vskip .2cm

i4 : IX = kernel(p)

\vskip .1cm

o4 = ideal | ae2-bcf aef-bdg af2-bdh aeg-bch cf2-deg fg-eh a2eh-bcdi 
\hfill \break
\phantom{dadadada} \qquad\qquad
afh-dei ag2-cei dg2-cfh agh-cfi bgh-e2i ah2-dgi bh2-efi |

o4 : Ideal of S

\vskip .1cm


i5 : degree(IX)

o5 = 18

\vskip .1cm

i6 : $\{$codim IX, pdim coker gens IX$\}$

o6 = $\{$4, 4$\}$

\quad \phantom{dada} 
}

\noindent This checks that the ideal {\tt IX}
has codimension $4$ and degree $18$ and is 
Cohen-Macaulay. We now compute its initial ideal with
respect to the reverse lexicographic term order:

\vskip .3cm

{ \baselineskip=12pt \tt


i7 : inIX = ideal leadTerm gens gb IX

\vskip .1cm

o7 = ideal | fg bh2 ah2 bgh agh afh dg2 ag2 aeg cf2 af2 aef ae2 a2eh |

o7 : Ideal of S

\vskip .2cm

i8 : $\{$codim inIX, pdim coker gens inIX$\}$

o8 = $\{$4, 5$\}$

}

\noindent Thus the initial monomial ideal {\tt inIX} is not Cohen-Macaulay.

It was conjectured in [23, page 137, line 14] that this ideal
provides a negative answer to Question 4.1, i.e., that \underbar{all}
of its initial monomial ideals fail to be Cohen-Macaulay.
Unfortunately, this is incorrect, as the following change
of variable order  demonstrates:

\vskip .3cm

{ \baselineskip=12pt \tt
i1 : S = QQ[f,g,h,i,a,b,c,d,e];

\vskip .1cm

i2 : IX =  ideal(a*e\^{$\,\!$}2-b*c*f, \ a*e*f-b*d*g,
             $ \ldots \ldots\, $, b*h\^{$\,\!$}2-e*f*i);


\vskip .2cm

i3 : inIX = ideal leadTerm gens gb IX

\vskip .1cm

o3 = ideal | fg gbd hbc fbc fhc f2c h2b ghb h2a gha fha 

\qquad\qquad\qquad\qquad g2a f2a g3d ibcd |

o3 : Ideal of S

\vskip .2cm

i4 : $\{$ codim inIX, pdim coker gens inIX $\}$

o4 = $\{$ 4, 4 $\}$

}

\vskip .1cm

\noindent 
In this new term order the initial monomial ideal 
{\tt inIX} is Cohen-Macaulay.
We conclude that Question 4.1 remains an open problem,
for  the time being.


\vskip 3.2cm

\noindent {\bf  Acknowledgements. }
I wish to thank Michael Stillman and Burt Totaro for
helpful communications.
This project was partially supported by a David and Lucile Packard
Fellowship and an NSF National Young Investigator Fellowship.
It was done while the author held a visiting position
at the Research Institute for Mathematical Sciences of
Kyoto University. Many thanks to my Japanese friends
for a terrific year 97/98 in Kyoto.


\vskip 2.5cm

{\baselineskip=14pt

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\vskip .1cm

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}

\bye

